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310 X EXISTENCE THEOREMS

and we therefore consider the difference Atf) = <7(0 - £U(x₀ + &, v(£, z)) • Sj. =

A'(0 =

' (z, *,) - &*(0 ' (#(£) • z,

using the relation D^f , z) = f/(x₀ + f z, K£ z)) • z = 5(0 - z. But relation
(10.9.4.1) yields in particular

(z, *) + ^(0 • (5(0 ' z, *) = C(0 • &, z) + A(Q • (5(£) • *, z)

hence

s_l9 z) = ^(Q • (A«), z).

Furthermore, A(0) = 0; but the only solution of the linear differential equa-
tion r' = ^(Q-(r,z) which vanishes for £ = 0 is r(£) = 0 (10.6.3), hence
= 0 for |£| < 2, which proves the relation

D₂ 1?(^ z) • st = £U(x₀ + &₉ v(£, z)) • 5¹!

for any _Sl e E, i.e. D₂ »({, z) = £U(x_Q + ^z, i?({, z)). This holds for |{| < 2
and Ijzli < j?/2M; in particular, for £ = 1, and putting x = x₀ + z, we obtain
_M'(jt) = t/(jc, n(jc)) for \\x - JCoil < J8/2M, which ends the proof.

(10.9.5) Suppose U is continuously differentiable in A x B // K = R_? twice
continuously differentiable if K = C, and verifies the Frobenius condition
(10.9.4.1). Then, for each point (a, b) 6 A x B, there is an open ball S c A 0/
center a and an open ball T c B o/ center h, having the following properties:
(1) for any point (x₀ , y₀) e S x T, f/zere w ^z unique solution x -» w(x, x₀ , J₀) o/
(10.9.1), defined in S #/?d jwc/z //za/ w(x₀ , JC_Q , j₀) = j>₀; (2) u is continuously
differentiable in S x S x T. If in addition E and F tfre finite dimensional and
U is p times continuously differentiable (resp. indefinitely differentiable, resp.
analytic) in A x B, then u is p times continuously differentiable (resp. in-
definitely differentiable, resp. analytic) in S x S x T.

Finally, there is an open ball W c T of center b such that, for every point
(x, x₀ , jo) e S x S x W, //i<? equation y₀ = w(x₀ , x, y) has a unique solution

Let S₀ c A be an open ball of center a and radius a, T₀ c B an open
ball of center 6 and radius & such that || J7(jc, j>)|| < M in S₀ x T₀ . Consider
the ordinary differential equation

(1 0.9.5.1) H/ = U(x₀ 4- fe J₀ + w) - z =

_f z,



	310 X EXISTENCE THEOREMS and we therefore consider the difference Atf) = <7(0 - £U(x₀ + &, v(£, z)) • Sj. =

	A'(0 = ' (z, ,) - &(0 ' (#(£) • z,

	using the relation D^f , z) = f/(x₀ + f z, K£ z)) • z = 5(0 - z. But relation (10.9.4.1) yields in particular

	(z, ) + ^(0 • (5(0 ' z, ) = C(0 • &, z) + A(Q • (5(£) • , z) hence s_l9 z)* = ^(Q • (A«), z).

	Furthermore, A(0) = 0; but the only solution of the linear differential equa- tion r' = ^(Q-(r,z) which vanishes for £ = 0 is r(£) = 0 (10.6.3), hence = 0 for \|£\| < 2, which proves the relation D₂ 1?(^ z) • st = £U(x₀ + &₉ v(£, z)) • 5¹!

	for any _Sl e E, i.e. D₂ »({, z) = £U(x_Q + ^z, i?({, z)). This holds for \|{\| < 2 and Ijzli < j?/2M; in particular, for £ = 1, and putting x = x₀ + z, we obtain _M'(jt) = t/(jc, n(jc)) for \\x - JCoil < J8/2M, which ends the proof.

	(10.9.5) Suppose U is continuously differentiable in A x B // K = R_? twice continuously differentiable if K = C, and verifies the Frobenius condition (10.9.4.1). Then, for each point (a, b) 6 A x B, there is an open ball S c A 0/ center a and an open ball T c B o/ center h, having the following properties: (1) for any point (x₀ , y₀) e S x T, f/zere w ^z unique solution x -» w(x, x₀ , J₀) o/ (10.9.1), defined in S #/?d jwc/z //za/ w(x₀ , JC_Q , j₀) = j>₀; (2) u is continuously differentiable in S x S x T. If in addition E and F tfre finite dimensional and U is p times continuously differentiable (resp. indefinitely differentiable, resp. analytic) in A x B, then u is p times continuously differentiable (resp. in- definitely differentiable, resp. analytic) in S x S x T. Finally, there is an open ball W c T of center b such that, for every point (x, x₀ , jo) e S x S x W, //i<? equation y₀ = w(x₀ , x, y) has a unique solution

	Let S₀ c A be an open ball of center a and radius a, T₀ c B an open ball of center 6 and radius & such that \|\| J7(jc, j>)\|\| < M in S₀ x T₀ . Consider the ordinary differential equation

	(1 0.9.5.1) H/ = U(x₀ 4- fe J₀ + w) - z =	_f z,